Torsion and Positive Rank in an Elliptic Family Arising from Cubic 2-Cycles
Abstract
We study the elliptic family Et:Y2=X3+4X2+16t2 arising from rational $2$-cycles of x3+bx+a. For every t∈ℚ×, we prove that Pt=(0,−4t) has infinite order, so every nonsingular rational fiber has positive rank. We determine its rational torsion subgroup: it is cyclic of order $2$ precisely when t=u(u2+4)/4 for some u∈ℚ×, and is trivial otherwise. No such fiber admits a rational $3$- or $5$-isogeny. An explicit birational dictionary then shows that, for every fixed a∈ℚ×, infinitely many b∈ℚ yield a rational $2$-cycle of x3+bx+a. The uniform non-torsion assertion follows from Nagell--Lutz integrality. The torsion exclusions combine elementary $2$-descent with explicit genus-$3$ curves, an unconditional rank-zero Prym argument, and two-cover descent with elliptic Chabauty. Exact Magma and SageMath certificates accompany the computer-assisted steps.
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BibTeX
@misc{chuysurichay2026torsion,
title = {{Torsion and Positive Rank in an Elliptic Family Arising from Cubic 2-Cycles}},
author = {Sompong Chuysurichay and Sawian Jaidee and Chatchawan Panraksa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.15109},
eprint = {2606.15109},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.15109},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Torsion and Positive Rank in an Elliptic Family Arising from Cubic 2-Cycles
AU - Sompong Chuysurichay
AU - Sawian Jaidee
AU - Chatchawan Panraksa
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2606.15109
UR - https://arxiv.org/abs/2606.15109
AB - We study the elliptic family E_t:Y^2=X^3+4X^2+16t^2 arising from rational $2$-cycles of x^3+bx+a. For every t∈ℚ^×, we prove that P_t=(0,−4t) has infinite order, so every nonsingular rational fiber has positive rank. We determine its rational torsion subgroup: it is cyclic of order $2$ precisely when t=u(u^2+4)/4 for some u∈ℚ^×, and is trivial otherwise. No such fiber admits a rational $3$- or $5$-isogeny. An explicit birational dictionary then shows that, for every fixed a∈ℚ^×, infinitely many b∈ℚ yield a rational $2$-cycle of x^3+bx+a. The uniform non-torsion assertion follows from Nagell--Lutz integrality. The torsion exclusions combine elementary $2$-descent with explicit genus-$3$ curves, an unconditional rank-zero Prym argument, and two-cover descent with elliptic Chabauty. Exact Magma and SageMath certificates accompany the computer-assisted steps.
ER -